NAME

Quantize - ImageMagick’s color reduction algorithm.

SYNOPSIS

#include<magick.h>

DESCRIPTION

This document describes how ImageMagick performs color reduction on an
image. To fully understand this document, you should have a knowledge
of basic imaging techniques and the tree data structure and
terminology.
For purposes of color allocation, an image is a set of n pixels, where
each pixel is a point in RGB space. RGB space is a 3-dimensional
vector space, and each pixel, pi, is defined by an ordered triple of
red, green, and blue coordinates, (ri,gi,bi).
Each primary color component (red, green, or blue) represents an
intensity which varies linearly from 0 to a maximum value, cmax, which
corresponds to full saturation of that color. Color allocation is
defined over a domain consisting of the cube in RGB space with opposite
vertices at (0,0,0) and (cmax,cmax,cmax). ImageMagick requires cmax=255.
The algorithm maps this domain onto a tree in which each node
represents a cube within that domain. In the following discussion,
these cubes are defined by the coordinate of two opposite vertices: The
vertex nearest the origin in RGB space and the vertex farthest from the
origin.
The tree’s root node represents the the entire domain, (0,0,0) through
(cmax,cmax,cmax). Each lower level in the tree is generated by
subdividing one node’s cube into eight smaller cubes of equal size.
This corresponds to bisecting the parent cube with planes passing
through the midpoints of each edge.
The basic algorithm operates in three phases: Classification,Reduction, and Assignment. Classification builds a color description
tree for the image. Reduction collapses the tree until the number it
represents, at most, is the number of colors desired in the output
image. Assignment defines the output image’s color map and sets each
pixel’s color by reclassification in the reduced tree. Our goal is to
minimize the numerical discrepancies between the original colors and
quantized colors. To learn more about quantization error, see
MEASURING COLOR REDUCTION ERROR later in this document.
Classification begins by initializing a color description tree of
sufficient depth to represent each possible input color in a leaf.
However, it is impractical to generate a fully-formed color description
tree in the classification phase for realistic values of cmax. If
color components in the input image are quantized to k-bit precision,
so that cmax=2k-1, the tree would need k levels below the root node
to allow representing each possible input color in a leaf. This
becomes prohibitive because the tree’s total number of nodes is
Σ ki=1 8k
A complete tree would require 19,173,961 nodes for k=8,cmax=255.
Therefore, to avoid building a fully populated tree, ImageMagick: (1)
Initializes data structures for nodes only as they are needed; (2)
Chooses a maximum depth for the tree as a function of the desired
number of colors in the output image (currently log4(colormapsize)+2).
A tree of this depth generally allows the best representation of the
source image with the fastest computational speed and the least amount
of memory. However, the default depth is inappropriate for some
images. Therefore, the caller can request a specific tree depth.
For each pixel in the input image, classification scans downward from
the root of the color description tree. At each level of the tree, it
identifies the single node which represents a cube in RGB space
containing the pixel’s color. It updates the following data for each
such node:
n1: Number of pixels whose color is contained in the RGB cube which
this node represents;
n2: Number of pixels whose color is not represented in a node at
lower depth in the tree; initially, n2=0 for all nodes
except leaves of the tree.
Sr,Sg,Sb:
Sums of the red, green, and blue component values for all pixels
not classified at a lower depth. The combination of these sums
and n2 will ultimately characterize the mean color of a set of
pixels represented by this node.
E: The distance squared in RGB space between each pixel contained
within a node and the nodes’ center. This represents the
quantization error for a node.
Reduction repeatedly prunes the tree until the number of nodes with n2>0 is less than or equal to the maximum number of colors allowed in
the output image. On any given iteration over the tree, it selects
those nodes whose E value is minimal for pruning and merges their color
statistics upward. It uses a pruning threshold, Ep, to govern node
selection as follows:
Ep = 0
while number of nodes with (n2 > 0) > required maximum number of
colors
prune all nodes such that E <= Ep
Set Ep to minimum E in remaining nodes
This has the effect of minimizing any quantization error when merging
two nodes together.
When a node to be pruned has offspring, the pruning procedure invokes
itself recursively in order to prune the tree from the leaves upward.
The values of n2Sr,Sg, and Sb in a node being pruned are always
added to the corresponding data in that node’s parent. This retains
the pruned node’s color characteristics for later averaging.
For each node, n2 pixels exist for which that node represents the
smallest volume in RGB space containing those pixel’s colors. When n2>0 the node will uniquely define a color in the output image. At the
beginning of reduction, n2=0 for all nodes except the leaves of the
tree which represent colors present in the input image.
The other pixel count, n1, indicates the total number of colors within
the cubic volume which the node represents. This includes n1-n2
pixels whose colors should be defined by nodes at a lower level in the
tree.
Assignment generates the output image from the pruned tree. The output
image consists of two parts: (1) A color map, which is an array of
color descriptions (RGB triples) for each color present in the output
image; (2) A pixel array, which represents each pixel as an index into
the color map array.
First, the assignment phase makes one pass over the pruned color
description tree to establish the image’s color map. For each node
with n2>0, it divides Sr,Sg, and Sb by n2. This produces the mean
color of all pixels that classify no lower than this node. Each of
these colors becomes an entry in the color map.
Finally, the assignment phase reclassifies each pixel in the pruned
tree to identify the deepest node containing the pixel’s color. The
pixel’s value in the pixel array becomes the index of this node’s mean
color in the color map.
Empirical evidence suggests that distances in color spaces such as YUV,
or YIQ correspond to perceptual color differences more closely than do
distances in RGB space. These color spaces may give better results
when color reducing an image. Here the algorithm is as described
except each pixel is a point in the alternate color space. For
convenience, the color components are normalized to the range 0 to a
maximum value, cmax. The color reduction can then proceed as
described.

MEASURINGCOLORREDUCTIONERROR

Depending on the image, the color reduction error may be obvious or
invisible. Images with high spatial frequencies (such as hair or
grass) will show error much less than pictures with large smoothly
shaded areas (such as faces). This is because the high-frequency
contour edges introduced by the color reduction process are masked by
the high frequencies in the image.
To measure the difference between the original and color reduced images
(the total color reduction error), ImageMagick sums over all pixels in
an image the distance squared in RGB space between each original pixel
value and its color reduced value. ImageMagick prints several error
measurements including the mean error per pixel, the normalized mean
error, and the normalized maximum error.
The normalized error measurement can be used to compare images. In
general, the closer the mean error is to zero the more the quantized
image resembles the source image. Ideally, the error should be
perceptually-based, since the human eye is the final judge of
quantization quality.
These errors are measured and printed when -verbose and -colorsarespecifiedonthecommandline:meanerrorperpixel:
is the mean error for any single pixel in the image.
normalizedmeansquareerror:
is the normalized mean square quantization error for any single
pixel in the image.
This distance measure is normalized to a range between 0 and 1.
It is independent of the range of red, green, and blue values in
the image.
normalizedmaximumsquareerror:
is the largest normalized square quantization error for any
single pixel in the image.
This distance measure is normalized to a range between 0 and 1.
It is independent of the range of red, green, and blue values in
the image.

SEEALSO

COPYRIGHT

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ACKNOWLEDGEMENTS

Paul Raveling, USC Information Sciences Institute, for the original
idea of using space subdivision for the color reduction algorithm.
With Paul’s permission, this document is an adaptation from a document
he wrote.